For the tomato sauce, heat the oil and garlic in a large pan. When you begin to smell the garlic, add the tomatoes and a good pinch of salt and bring to the boil. Lower the heat and cook uncovered for about 20–25 minutes, until it has all merged into a sauce. Add the basil and 250 ml(9 fl oz/1 cup) of hot water towards the end of the cooking time. Purée until smooth, minus the garlic if you’d prefer.
To make the béchamel, melt the butter in a small saucepan. Whisk in the flour and cook for a few minutes, stirring constantly, then begin adding the warm milk. It will be immediately absorbed, so work quickly, whisking with one hand while adding ladlefuls of milk with the other. When the sauce seems to be smooth and not too stiff, add salt, pepper and a grating of nutmeg and continue cooking, even after it comes to the boil, for 5 minutes or so, mixing all the time. It should be a very thick and smooth sauce.
Preheat the oven to 180°C (350°F/Gas 4) and grease a deep 22 × 30 cm(8½ × 12 inch) baking dish. Drizzle some béchamel over the bottom of the dish to cover it very thinly. Put a slightly overlapping layer of lasagne sheets on top. Dollop a thin layer of tomato sauce over that, spreading it with the back of the ladle. Add about two ladlefuls of béchamel in long drizzles and then cover with a sprinkling of parmesan. Add another layer of lasagne sheets, then tomato, béchamel and parmesan as before, and then repeat the layers one more time. You should have about 3 tablespoons of tomato sauce and a good amount of béchamel left. Make a final layer of lasagne sheets and cover with all the remaining béchamel. Dollop the tomato sauce here and there and sprinkle with any remaining parmesan. Bake for about 30 minutes, or until it is bubbling and golden on top.